Spherical mean
In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.
Consider an open set U in the Euclidean space Rn and a continuous function u defined on U with real or complex values. Let x be a point in U and r > 0 be such that the closed ball B(x, r) of center x and radius r is contained in U. The spherical mean over the sphere of radius r centered at x is defined as
where ∂B(x, r) is the (n−1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and ωn−1(r) is the "surface area" of this (n−1)-sphere.
Equivalently, the spherical mean is given by
where ωn−1 is the area of the (n−1)-sphere of radius 1.
The spherical mean is often denoted as
The spherical mean is also defined for Riemannian manifolds in a natural manner.
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